We rely on density-functional theory for classical particles, a.k.a., classical-DFT [1]}, [2]}, generalized for the inclusion of a stationary temperature distribution. In particular, we aim to derive an asymptotic approximation for the free-energy density \(f(r)\)  of the fluid at the local temperature \(T(r)\) , written as a unique functional of \(\rho _{+/-}(r)\) . The ideal-gas contribution to \(f(r)\)  can be written exactly as \(f_\text{id}(r)=\sum _iT(r)\rho _i(r)\left[\ln \left(\Lambda ^3(r)\rho _i(r)\right)-1\right]\) , with \(\Lambda (r)\)  the local de Broglie wavelength [3]}. To derive an approximation of the excess contribution, we start separating a mean-field electrostatic term that expresses the self-interaction of the ionic charge-density \(\rho _Q(r)=Z(\rho _+(r)-\rho _-(r))\)  with itself, i.e., \(f_H(r)=\frac{1}{2} \rho _Q(r)\phi (r)\) , with \(\phi (r) = \int dr^{\prime } \rho _Q(r^{\prime })/|r^{\prime }-r|\)  the electrostatic potential. The remaining (neutralized) part of the excess free-energy density \(f^\text{exc}(r)\)  contains information about the short-range correlations in the fluid and its functional dependence from \(\rho _{+/-}(r)\)  is generally unknown. However, in the limit of small and slowly varying inhomogeneities, which are implied by the asymptotic response of the ions' densities far from an external perturbation, a rigorous expression for \(f^\text{exc}(r)\)  can be provided that is grounded on an asymptotic gradient expansion [4]}, [5]}. In particular, we rely on a square-gradient approximation (SGA) already adopted in the phenomenological treatment of non-equilibrium steady-states [6]}, [7]}:
\(f^\text{exc}_\text{SGA}(r) = f^\text{exc}_\text{LDA}(r) + \frac{1}{2}\sum _{ij} A^T_{ij} \nabla \rho _i(r)\cdot \nabla \rho _{j}(r)\, ,\) 
